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Theorem nfald 1753
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfald (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . . 4 𝑦𝜑
21nfri 1512 . . 3 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
42, 3alrimih 1462 . 2 (𝜑 → ∀𝑦𝑥𝜓)
5 nfnf1 1537 . . . 4 𝑥𝑥𝜓
65nfal 1569 . . 3 𝑥𝑦𝑥𝜓
7 hba1 1533 . . . 4 (∀𝑦𝑥𝜓 → ∀𝑦𝑦𝑥𝜓)
8 sp 1504 . . . . 5 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
98nfrd 1513 . . . 4 (∀𝑦𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
107, 9hbald 1484 . . 3 (∀𝑦𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
116, 10nfd 1516 . 2 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝑦𝜓)
124, 11syl 14 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1346  wnf 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-4 1503  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  dvelimALT  2003  dvelimfv  2004  nfeudv  2034  nfeqd  2327  nfraldw  2502  nfraldxy  2503  nfiotadw  5163  nfixpxy  6695  bdsepnft  13922
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